FRACTIONAL VECTOR AUTOREGRESSIVE MODELS FOR ENVIRONMENTAL DATA

Transcription

1 FRACTIONAL VECTOR AUTOREGRESSIVE MODELS FOR ENVIRONMENTAL DATA Mariagrazia Granturco Eugenia Nissi Dipartimento di Metodi Quantitativi e Teoria Economica Università degli Studi G. D Annunzio Viale Pindaro, Pescara m.granturco@dmqte.unich.it - nissi@dmqte.unich.it TIES 2002 Annual Conference Genova 8-22 giugno 2002

2 AIMS OF THE PAPER Time series of pollutants usually show long memory dependence In this setting, the aims of the paper is to analyze times series of the concentration of carbon monoxide in the Bergamo District by means of multivariate ARFIMA Models

3 Structure of the talk: ARFIMA Models: methodological issues VAR Models: methodological issues Proposed methodology for multiple time series: Estimation of the common long memory parameter; Estimation of VAR model on the residual short memory component Case of study

4 DEFINITIONS OF LONG MEMORY TIME PROCESSES! THE CORRELATION DECAY TO ZERO WITH AN HYPERBOLIC RATE n α + k lim ρ k c k ρ k con: α 0, c > 0! THE SPECTRAL DENSITY FUNCTION IS NOT FINITE AT ZERO FREQUENCY 2 σ + iλk α lim λ 0 f λ lim ρ k e cλ λ 0 2π k

5 TEMPORAL PROCESSES WITH LON MEMORY STATIONARY INCREMENTS OF SELF SIMILARITY PROCESS ARFIMA0,d,0 PROCESS Xt Yt-Yt-: Yt c -H Ytc Xt -B -d εt IF: IF:! H ½,: INVERTIBLE AND d 0,½:INVERTIBLE AND STATIONARY PROCESS WITH LONG STATIONARY PROCESS WITH LONG MEMORY MEMORY! Xt~Nµ,Σ: FRACTIONAL GAUSSIAN NOISE WITH SELF SIMILARITY PARAMETER H! Xt~Nµ,Σ: ARFIMA PROCESS WITH FRACTIONAL DIFFERENCE OF ORDER d ASIMPTOTICALLY EQUIVALENT PROCESSES: Hd+ ½ Generalization : ARFIMA PROCESSES p,d,q φb -B d Xt θbεt

6 TEST FOR LONG MEMORY! RESCALED RANGE TEST ˆ 2 2 ] [ min ] max [ R/S 2 2 t V q w S x x n x x x x d q x t n n k n k n k n k + γ! KPSS TEST 2 ˆ n d t d t n dr r V t x dr r V x x q n β α σ η CLASSIC R/S MODIFIED R/S H 0 x, x 2,..., x I0 H x, x 2,..., x I! RESCALED VARIANCE TEST ˆ,..., / 2 2 t V q n S S VAR S V d n n σ! LOBATO-ROBINSON TEST m m m k i m v N I I v m LB ln ln 0, ~ λ λ

7 ESTIMATION OF THE LONG MEMORY PARAMETER H ESTIMATION FROM CORRELOGRAM: Lim ρ k k c k 2H ESTIMATION FROM SAMPLE VARIANCE: 2H Lim VAR X c n n 2 [ ] ESTIMATION FROM R/S REGRESSION: Lim n [ Log R / S ] A + H Log n ESTIMATION FROM PERIODOGRAM: Lim f λ c λ 2H λ 0 GPH ESTIMATION: 2 σ fa 0 2 λ logi λ Log [ I λ ] Log d Log 4sin + 2π 2 log f X λ APPROXIMATE MLE WHITTLE: L * W n* I * λ n* n*log2π n*log log f λ, κ* n * n* f λ, κ*

8 SIMULATION RESULTS LOG-LOG Stime da CORRELOGRAM correlogramma ESTIMATES ESTIMATES BY SAMPLE VARIANCE Stime da varianza della media campionaria H0.6 H0.7 H0.8 H H0.6 H0.7 H0.8 H SQR BIAS MSE SQR BIAS MSE R/S Stime REGRESSION da regressione ESTIMATES R/S LOG-LOG PERIODOGRAM ESTIMATES Stime da periodogramma H0.6 H0.7 H0.8 H H0.6 H0.7 H0.8 H SQR BIAS MSE SQR BIAS MSE GPH Stime ESTIMATES GPH Stime WHITTLE di M.V. MLE di ESTIMATES Whittle H0.6 H0.7 H0.8 H H0.6 H0.7 H0.8 H SQR BIAS MSE SQR BIAS MSE

9 VAR MODEL OF ORDER p yt v + A yt-+ +Ap yt-p+εt t ±, ±2 yt: K multivariate random vector v: K vector of intercept terms A, Ap: K K matrices of coefficients εt: K-dimensional white noise process with non singular covariance matrix

10 CONCENTRATION OF CARBON MONOXIDE CO IN BERGAMO DISTRICT PONTE S.PIETRO SERIATE DALMINE S.GIORGIO GARIBALDI NEMBRO TREVIGLIO CISERANO COSTA VOLPINO GOISIS

11 LOGARITM OF DATA PONTE S.PIETRO NEMBRO SERIATE TREVIGLIO DALMINE CISERANO S.GIORGIO COSTA VOLPINO GARIBALDI GOISIS

12 RAW TIME SERIES CORRELATIONS Ponte S.Pietro Nembro Seriate Treviglio Dalmine Ciserano S. Giorgio Costa Volpino Garibaldi Ponte S.Pietro Nembro Seriate Treviglio Dalmine Ciserano S. Giorgio Costa Volpino Garibaldi Goisis Goisis PERIODOGRAM CORRELATIONS Ponte S.Pietro Nembro Seriate Treviglio Dalmine Ciserano S. Giorgio Costa Volpino Garibaldi Goisis Ponte S.Pietro Nembro Seriate Treviglio Dalmine Ciserano S. Giorgio Costa Volpino Garibaldi Goisis

13 SEASONAL COMPONENT OF LOG DATA NEMBRO SERIATE TREVIGLIO DALMINE CISERANO S.GIORGIO GARIBALDI Log Series Seasonal Component

14 ACF OF DESEASONALISED LOG-SERIES NEMBRO SERIATE TREVIGLIO DALMINE CISERANO S.GIORGIO GARIBALDI

15 PACF OF DESEASONALISED LOG-SERIES NEMBRO SERIATE TREVIGLIO DALMINE CISERANO S.GIORGIO GARIBALDI

16 PERIODOGRAM OF DESEASONALISED LOG-SERIES NEMBRO SERIATE TREVIGLIO DALMINE CISERANO S.GIORGIO GARIBALDI

17 TEST FOR LONG MEMORY Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob serie V/S R/S KPSS_trend KPSS_mu Lob-Rob Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob Serie V/S R/S KPSS_trend KPSS_mu Lob-Rob : Truncation lag of correlogram 2: Critical value V/S test : Critical value R/S test.747 4: Critical value KPSS test : Critical value KPSS2 test : Critical value LR test.96 7: Truncation frequency of periodogram

18

19 H0.69

20 H0.70

21 VAR MODEL ON THE RESIDUAL SHORT MEMORY COMPONENT Nembro Lag- Seriate Lag- Treviglio Lag- Dalmine Lag- Ciserano Lag- S. Giorgio Lag- Garibaldi Lag- Nembro Seriate Treviglio Dalmine Ciserano S. Giorgio Garibaldi

22 NON SIGNIFICANT ACF COEFFICIENTS FOR RESIDUALS FROM VAR MODEL

23 GOODNESS OF FIT FOR ESTIMATED MODELS FIT VAR ON LOG DATA

24 SCATTER PLOT FOR ESTIMATED MODELS

25 FORECASTING FROM ESTIMATED MODELS st period forecasting 2nd period forecasting 3th period forecasting th period forecasting 5th period forecasting 6th period forecasting th period forecasting Observed data GPH-VAR forecasting VAR forecasting Periodogram-VAR forecasting